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Effect of U on the Electronic Properties of Neodymium Gallate (NdGaO3): Theoretical and Experimental Studies Ali Hussain Reshak,*,† M. Piasecki,‡ S. Auluck,§ I. V. Kityk,| R. Khenata,⊥ B. Andriyevsky,# C. Cobet,∇ N. Esser,∇ A. Majchrowski,O M. S´wirkowicz,[ R. Diduszko,[ and W. Szyrski[ Institute of Physical Biology, South Bohemia UniVersity, NoVe Hrady 37333, Czech Republic, Institute of Physics, J.Dlugosz UniVersity, Al. Armii Krajowej13/15, 42-200 Czestochowa, Poland, Physics Department, Indian Institute of Technology Kanpur, Kanpur (UP) 208016, India, Electrical Engineering Department, Technological UniVersity of Czestochowa, Al.Armii Krajowej 17/19, 42-200 Czestochowa, Poland, Laboratoire de Physique Quantique et de Mode´lisation Mathe´matique (LPQ3M), De´partement de Technologie, UniVersite´ de Mascara, 29000-Algeria, Faculty of Electronics and Computer Sciences, Koszalin UniVersity of Technology, 2 Sniadeckich St, PL-75-453 Koszalin, Poland, Institute for Analytical Sciences, Department Berlin, 9 Albert-Einstein-Strasse, D-12489 Berlin, Germany, Institute of Applied Physics, Military UniVersity of Technology, Kaliskiego 2, 00-908 Warsaw, Poland, and Institute of Electronic Materials Technology, Wolczynska 133, 01-919 Warsaw, Poland ReceiVed: August 16, 2009; ReVised Manuscript ReceiVed: September 12, 2009
We have performed a density functional calculation for the centrosymmetric neodymium gallate using a fullpotential linear augmented plane wave method with the LDA and LDA+U exchange correlation. In particular, we explored the influence of U on the band dispersion and optical transitions. Our calculations show that U ) 0.55 Ry gives the best agreement with our ellipsometry data taken in the VUV spectral range with a synchrotron source. Our LDA+U (U ) 0.55) calculation shows that the valence band maximum (VBM) is located at T and the conduction band minimum (CBM) is located at the center of the Brillouin zone, resulting in a wide indirect energy band gap of about 3.8 eV in excellent agreement with our experiment. The partial density of states show that the upper valence band originates predominantly from Nd-f and O-p states, with a small admixture of Nd-s/p and Ga-p B-p states, while the lower conduction band prevailingly originates from the Nd-f and Nd-d terms with a small contribution of O-p-Ga-s/p states. The Nd-f states in the upper valence band and lower conduction band have a significant influence on the energy band gap dispersion which is illustrated by our calculations. The calculated frequency dependent optical properties show a small positive uniaxial anisotropy. 1. Introduction The rare-earth oxide compounds with perovskite structure have very interesting properties which are very useful in solid state physics.1 They have received much attention in connection with the discovery of high TC superconductivity and magnetoresistance,1 metal-insulator MI transition,2 and IR thermochromic properties.3,4 Among these, the rare-earth oxide compound neodymium gallate (NdGaO3) has been studied by numerous workers. The first communication on structure of neodymium gallate (NdGaO3) was reported by Geller.5 Brusset et al.6 studied the crystal structure in more detail using X-ray diffraction. Previous studies indicate that neodymium gallate is widely used as a substrate material for high temperature superconductors, GaN film deposition, and colossal magnetoresistive (CMR) film epitaxy.7-10 Vasylechko et al.10 show that the interest in this material stems mainly from its technically attractive dielectric properties (low ε and tan δ) and the low * Corresponding author. Phone: +420 777729583. E-mail: maalidph@ yahoo.co.uk. † South Bohemia University. ‡ J.Dlugosz University. § Indian Institute of Technology Kanpur. | Technological University of Czestochowa. ⊥ Universite´ de Mascara. # Koszalin University of Technology. ∇ Institute for Analytical Sciences. O Military University of Technology. [ Institute of Electronic Materials Technology.
mismatch between film and substrate lattice parameters, both at the deposition and application temperatures. Senyshyn et al.11 reported the suitability of the neodymium gallate for high temperature superconductors CMR and GaN substrate materials and as interconnecting material in solid oxide fuel cells which operate at 77 and 1000 K. Thus, the thermal behavior of this compound turns out to be an important object for research. Recently, Galicka et al.12 have reported the electronic structure of NdNiO3/NdGaO3 thin films with various thicknesses using photoemission spectroscopy at 300 and 169 K. They reported that the XPS results are consistent with the literature ab initio calculations of the NdNiO3 electronic structure. We have not come across any work either experimental or theoretical on the optical functions or first principles electronic structure of neodymium gallate in the literature. A detailed description of the electronic and spectral features of the optical properties of neodymium gallate would bring us important insights in understanding the origin of the electronic band structure and densities of states. We report measurements of the frequency dependent dielectric functions performed at BESSY II and in our laboratory using UV-vis spectroscopy in the energy range 2.0-15.0 eV. The present study is also aimed toward calculations using the full-potential linear augmented plane wave (FP-LAPW) method which has proven to be a very accurate method13,14 for the computation of the electronic structure of solids within a framework of density functional theory (DFT). We have performed calculations at two levels of
10.1021/jp908025p CCC: $40.75 2009 American Chemical Society Published on Web 10/26/2009
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approximations: the local density approximation (LDA) and the Coulomb corrected LDA+U. In section 2, we present the experimental procedure. The theoretical aspects of the calculation methods were presented in section 3. The calculated band structure and density of states are given in sections 4a and 4b, and devoted to the optical properties. 2. Experimental Procedure 2.1. Crystal Growth. Neodymium gallate (NdGaO3) single crystals melt congruently, so they can be grown by the Czochralski method from the stoichiometric melts. The charge material was prepared on the basis of 99.998% purity Ga2O3 (Auer Remy), 5N grade Nd2O3 (Metal Rare Earth Ltd., China). The reagents were mixed and then heated in a resistance furnace at 1150 °C for 20 h to complete the synthesis in the solid state. After that, the charge material was pressed isostatically. The growing atmosphere was nitrogen with 0.5 vol % of oxygen. The following conditions of the growth process have been applied: growth rate, 0.8-1.5 mm/h; rotation rate, 15-30 rpm; cooling after growth, at least 24 h. NdGaO3 single crystals were grown from an iridium crucible, 50 mm in height and diameter. Two active iridium heaters were placed below and above the crucible to diminish thermal gradients in the furnace. The system was placed in the Oxypuller 05-03 equipment. Inductive heating with the Hu¨ttinger generator was used. The as-grown single crystals were 20-22 mm in diameter and up to 60 mm in length. 2.2. Optical Measurements. We have measured the pseudodielectric function 〈ε〉 ) 〈ε1〉 + i〈ε2〉 of NdGaO3 crystals by spectroscopic ellipsometry using the synchrotron ellipsometer115,16 attached to the 3m Normal Incidence Monochromator (NIM) of the Berlin Electron Storage ring for Synchrotron radiation BESSY II in the photon energy range 2.0-15.0 eV with a resolution of 0.02 eV. The resolution of the 3m NIM is equal to R ) λ/δλ ) 50000 (λ is a wavelength). The angle of incidence was about 68° below and 45° above 10 eV, while the polarization of the incident beam was chosen 20° tilted with respect to the plane of incidence during the measurements. Below 10 eV a MgF2 polarizer and a rotating analyzer,
Figure 2. Calculated band structure: (a) LDA; (b) LDA+U.
Figure 1. Calculated crystal structure.
respectively, ensured a 99.998% degree of polarization. Above 10 eV triple reflection type polarizers are used, which provide about 99% polarization and guarantee a proper reduction of second order light from the monochromator. The sample’s surfaces were polished with surface roughness within 2 µm following profilometer data. We have not found any significant changes of the spectra in the heating and cooling regime. The statistics were done in order to achieve a χ(2) reproducibility of the data up to 0.002. 3. Theoretical Calculation NdGaO3, a ferroelastic crystal, has an orthorhombic space 16 ), with unit cell parameters a ) 0.5426 nm, b group Pbmn (D2h ) 0.5496 nm, and c ) 0.7707 nm and a highly complicated structure composed of 20 atoms per unit cell. In Figure 1, we show the crystal structure. We use a full-potential linear augmented plane wave method within density functional theory (DFT),17 as implemented in the package WIEN2k code.18
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Figure 3. (a) Calculated total density of states (states/eV unit cell) using LDA. (b) Calculated total densities of states (states/eV unit cell) using LDA+U. (c) Dispersion of the states around Fermi energy showing the influence of varying U on the value of the energy gap. (d) Calculated density of states (states/eV unit cell) LDA+U (U ) 0.55 Ry). (e-h) Calculated partial densities of states (states/eV unit cell) using LDA+U.
Keeping the lattice parameters fixed at the experimental values, we have optimized the structure by minimization of the forces (1 mRy/au) acting on the atoms. From the relaxed geometry, the electronic structure and the chemical bonding can be determined and various spectroscopes can be simulated and compared with experiment. Once the forces are minimized, we
can then find the charge density at these positions by turning off the relaxations and driving the system to self-consistency. The exchange and correlation potential used is the local density approximation (LDA). We find that LDA gives delocalized Nd-f at Fermi energy, making this compound metallic, which is in disagreement with our experiment which shows that NdGaO3
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is a semiconductor with an indirect energy band gap of about 3.8 eV. For oxides and other highly correlated compounds, LDA and GGA are known to fail to give the correct ground state. In these systems, the electrons are highly localized. The Coulomb repulsion between the electrons in open shells should be taken into account. As there is no exchange correlation functional that can include this in an orbital independent way, a simpler approach is to add the Hubbard-like on-site repulsion to the Kohn-Sham Hamiltonian. This is known as a LDA+U calculation. There are different ways in which this can be implemented. In our work, we have used the method of Anisimov et al.19 and Liechtenstein et al.20 where the Coulomb (U) and exchange (J) parameters are used. We have performed calculations using the LDA+U method. As there are no experimental measurements of U for this compound, we have decided to perform calculations for various values of U (0.15, 0.35, 0.55, 0.75, and 0.95 Ry), and keeping the J parameter fixed at 0.072 Ry. The main motivation for taking different U values is to find which U value gives the best fit to the measured energy gap. We take the full relativistic effects for core states and use the scalar relativistic approximation for the valence states. The muffin tin radii are taken to be 2.4 au (atomic units) for Nd atom and 1.85 au for Ga and O atoms. We used the parameters Kmax ) 9/RMT and lmax ) 10. The self-consistency was achieved by using 300 k-points in the irreducible Brillouin zone (IBZ). The density of states and the optical properties are calculated using 500 k-points of the IBZ. The self-consistent interactions were performed until convergence only when the integrated charge distance per formula unit, ∫|Fn - Fn-1| dr, between the input charge density [Fn-1(r)] and the output [Fn(r)], is less than 0.0001. 4. Results and Discussion a. Band Structure and Density of States. The band structure and total density of states using LDA and LDA+U are shown in Figures 2 and 3a. It is clear that LDA gives delocalized Nd-f states at Fermi energy, making NdGaO3 metallic, which is in disagreement with our experiment. In order to overcome this disagreement, we have performed calculations using LDA+U. Figure 3b and c shows the dramatic change in the density of states as a function of U. We notice that when using a small value of U ) 0.15 Ry the density of states show a gap opening around EF, making the compound a semiconductor with a small energy gap of about 1.1 eV. On increasing U from 0.35 to 0.95 Ry, we notice that the energy gap increases from 2.6 to 4.4 eV (see Figure 3c). We noticed that U ) 0.55 Ry gives an energy gap in agreement with the experimental value of 3.8 eV (see Figure 3d). Henceforth, we will discuss the results for this value of U. The band structure and total density of states (DOS) along with the Nd-s/p/d/f, Ga-s/p/d, and O-s/p partial DOS are shown in Figures 2 and 3d-h. Our calculations show that the valence band maximum (VBM) is located at T and the conduction band minimum (CBM) at the Γ point of the BZ, resulting in an indirect energy band gap of 3.8 eV. The band structure (and hence the DOS) could be divided into five groups/structures separated by gaps. From the partial DOS, we are able to identify the angular momentum character of the various structures. The lowest energy group around -15.0 eV has significant contributions from Nd-p and O-s states with a small admixture of Gas/p states. The second group around -12.5 eV is mainly from Ga-d states with small contributions of O-s/p, Nd-p, and Gas/p states. The group from -8.0 eV up to Fermi energy (EF) originates from Nd-f and O-p states with an admixture of Ndp/d, Ga-s/p, Nd-s, and O-s states. The bands from 3.8 eV (the
Reshak et al. fundamental energy gap) and above originate from Nd-f and Nd-d with an admixture of Ga-s/p, Nd-s/p, and O-s/p states. The electronic structure of the upper valence band originates predominantly from the Nd-f and O-p states, with a small contribution of Nd-s/p and Ga-p states, while the lower conduction band is dominated by Nd-f and Nd-d with a small contribution of O-p, Ga-s, and Ga-p states (see Figure 3e-h). The existence of Nd-f states in the upper valence band and lower conduction band has a significant effect on the energy band gap dispersion. From partial densities of states, one can see that there exists a strong hybridization between Ga-s and Nd-s and Nd-d and Nd-p in the energy range from -5.0 eV up to EF. At around -5.0 eV, Ga-p hybridized with Ga-s. The nature of chemical bonding can be elucidated from the total and partial densities of states. We find that the densities of states, extending from -8.0 eV up to Fermi energy (EF), are mainly from Nd-f (5.5 electrons/eV), O-p (1.1 electrons/eV), Nd-d (0.1 electrons/ eV), Ga-p (0.09 electrons/eV), and Nd-p states (0.08 electrons/ eV). This is obtained by comparing the total densities of states with the angular momentum projected densities of states of Ndp/d/f, O-p, and Ga-p states, as shown in Figure 2. These results show that some electrons from Nd-p/d/f, O-p, and Ga-p states are transferred into valence bands (VBs) and contribute in weak covalence interactions between O-O, Nd-Nd, and Ga-Ga atoms, and the substantial covalence interactions between Nd and O, Nd and Ga, and Ga and O atoms. Accordingly, we can also say that the covalent strength of Nd-O, Nd-Ga, and Ga-O bonds is weaker than that of O-O, Nd-Nd, and Ga-Ga bonds. b. First Order (Linear) Optical Susceptibilities. The investigated NdGaO3 crystal belongs to the orthorhombic space group Pbmn (D16 2h). It will therefore have three diagonal components of the dielectric tensor. Since the two lattice constants in the basal plane are almost equal, the dielectric tensor is the same as if it were a tetragonal structure. Hence, only two components of these dielectric tensors are important. These frequency-dependent dizz electric functions are εxx 2 (ω) and ε2 (ω) corresponding to the electric field direction perpendicular and parallel to the crystallographic c axis. The calculation of these frequency dependent dielectric functions requires the precise values of energy eigenvalues and electron wave functions. These are natural outputs of a band structure calculation. To calculate the direct interband contributions to the imaginary part of the frequency-dependent dielectric function, it is necessary to perform summation over the BZ structure for all possible transitions from the occupied to the unoccupied states. Taking the appropriate transition dipole matrix elements into account, we calculated the imaginary part of the frequencydependent dielectric functions using the expressions in refs 21 and 22.
εzz 2 (ω) )
εxx 2 (ω) )
6 mω2
12 mω2
∫BZ ∑
∫BZ ∑
Z |Pnn' (k)| 2 dSk ∇ωnn'(k)
(1)
X Y [|Pnn' (k)| 2 + |Pnn' (k)| 2] dSk ∇ωnn'(k)
(2) The above expressions are written in atomic units with e2 ) 1/m X ) 2 and p ) 1. ω is the photon energy, Pnn (k) is the x component of the transition dipole matrix elements between initial |nk〉 and final |n′k〉 band states with their eigenvalues En(k) and Εn′(k), respectively. ωnn′(k) is the energy difference ωnn′(k) ) En(k) - Εn′(k) and Sk is a constant energy surface Sk ) {k;ωnn′(k) ) ω}.
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xx zz Figure 4. (a) Calculated εxx 2 (ω) (dark solid curve) and ε2 (ω) (light dashed curve) spectra using LDA. (b) Calculated ε2 (ω) (dark solid curve) and exp xx εzz 2 (ω) (light dashed curve) spectra using LDA+U (U ) 0.55), along with ε2 (ω) (light solid curve). (c) Calculated ε1 (ω) (dark solid curve) and xx zz εzz 1 (ω) (light dashed curve) using LDA. (d) Calculated ε1 (ω) (dark solid curve) and ε1 (ω) (light dashed curve) using LDA+U (U ) 0.55), along xx zz with εexp 1 (ω) (light solid curve). (e) Calculated R (ω) (dark solid curve) and R (ω) (light dashed curve) spectrum using LDA+U (U ) 0.55). (f) Calculated refractive indices nxx(ω) (dark solid curve) and nzz(ω) (light dashed curve) spectrum using LDA+U (U ) 0.55).
Figure 4a and b depicts the variation of the imaginary part of the frequency dependent dielectric function. The spectral broadening is taken to be 0.1 eV which is typical for the experimental accuracy adopted for dielectric crystals. We first calculated the frequency-dependent dielectric functions using LDA. In LDA, this compound is metallic. Thus, the frequency dependent dielectric function contains the intraband transitions (significant for energies less than 1.0 eV) along with the interband transitions in disagreement with our measured frequency dependent dielectric functions. However, the LDA+U calculations give the correct semiconducting ground state. There is a significant difference in the imaginary part of the frequency dependent dielectric functions calculated with LDA (Figure 3a) and LDA+U (Figure 3b). The LDA calculation gives structures at around 3.5 and 7.0 eV, while the LDA+U gives only the 7.0 eV structure. The LDA+U calculation is in agreement with our optical measurements which also yield one main structure at around 7.0 eV. It is not hard to explain the difference in the two calculations. Following our calculated band structure and
DOS, for simplicity, we can call the bands Ga-s/p, Nd-d, and O-p which are above EF as (a), the bands Nd-f, around EF as (b), and (c) for the bands O-p and Ga-s/p which are below EF for LDA, while for LDA+U as (d) for the bands Nd-f, Ga-s/p, Nd-d, and O-p which are situated above EF, and (e) for Nd-f, O-p, and Ga-s/p bands which are below EF. Thus, in LDA, we have the transitions c-b and b-a occurring at 3.5 eV, while the c-a transition occurs at 7.0 eV. In LDA+U, we have only e-d transitions which occur at 7.0 eV. We have compared our calculated ε2xx(ω) and ε2zz(ω) with our measured ε2(ω) in Figure 3b. Good agreement is found in the matter of the critical point (threshold of the optical transitions) and the peak position. From Figure 3b, one can conclude that the critical point occurs at 3.8 eV for both ε2xx(ω) and ε2zz(ω) in excellent agreement with our experiment. The real parts ε1xx(ω) and ε1zz(ω) can be calculated from the spectral dependences of imaginary parts of the dielectric function ε2xx(ω) and ε2zz(ω) using Kramers-Kronig relations23 for LDA and LDA+U. These are shown in Figure 4c and d. There is
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zz weak anisotropy between εxx 1 (ω) and ε1 (ω). We have compared xx zz our calculated ε1 (ω) and ε1 (ω) with our measured ε1(ω). Good agreement is found. The static dielectric constant ε1(0) is given by the low energy limit of ε1(ω). Note that we do not include phonon contributions to the dielectric screening. ε1(0) corresponds to the static optical dielectric constant (ε∞). The zz calculated value of εxx 1 (0) is 4.025 and 4.15 for ε1 (0), indicating a small positive uniaxial anisotropy.24 In addition, we have calculated the spectral dependences for reflectivity spectraR(ω), and refractive indices n(ω). Figure 4e shows the calculated reflectivity spectra. We notice that a reflectivity maximum around 7.5 eV arises from interband transitions. A reflectivity minimum at energies ranging from about 11.0 eV confirms the occurrence of a collective plasma resonance. The depth of the plasma minimum is determined by the imaginary part of the dielectric function at the plasma resonance and is representative of the degree of overlap between the interband absorption regions. We note that at high energies (at around 13.5 eV) this compound shows a rapidly increasing reflectivity. The calculated refractive indices n(ω) show very week anisotropy, resulting in a very small birefringence, as illustrated in Figure 4f. As there are no experimental or theoretical results for the spectral features of the optical properties available for these compounds, we hope that our work will stimulate more works.
4. Conclusions LDA calculations give a metallic ground state, while the LDA+U calculations give the correct semiconducting ground state. We have adjusted the value of U to 0.55 Ry so as to get the experimental energy gap of 3.8 eV. It would be interesting if this finding could be corroborated by other experiments. The calculated optical properties from LDA+U are in agreement with our own measurements which yield one main structure at around 7.0 eV, while the LDA calculations give two structures at 3.5 and 7.0 eV. We have reported calculations of the frequency dependent dielectric function, reflectivity spectra, and refractive indices. The linear optical properties are found to be slightly anisotropic with small positive uniaxial anisotropy. Acknowledgment. This work was supported by the European Community - Research Infrastructure Action under the FP6 “Structuring the European Research Area” Programme through the Integrated Infrastructure Initiative “Integrating Activity on Synchrotron and Free Electron Laser Science” under the contract R II 3-CT-2004-506008 (1), by European Community through the “NanoCharM” project under the Seventh Framework Pro-
Reshak et al. gramme (FP7), and by the German Federal Ministry of Education and Research (BMBF) under the contract number 05 KS4KTB/3. We also want to thank the BESSY GmbH for help and support (project BESSY-09.1.81116). A.H.R. is thankful for support from the institutional research concept of the Institute of Physical Biology, UFB (No.MSM6007665808). References and Notes (1) Luis, F.; Kuz’min, M. D.; Bartolome, F.; Orera, V. M.; Bartolome, J. Phys. ReV. B 1998, 58, 798. (2) Lacorre, P.; Torrance, J. B.; Pannetier, J.; Nazzal, A. I.; Wang, P. W.; Huang, T. C. J. Solid State Chem. 1991, 91, 255. (3) Katsufuji, T.; Okimoto, Y.; Arima, T.; Tokura, Y. Phys. ReV. B 1995, 51, 4830. (4) Capon, F.; Laffez, P.; Bardeau, J.-F.; Simon, P.; Lacorre, P. Appl. Phys. Lett. 2002, 81, 619. (5) Geller, S. Acta Crystallogr. 1957, 10, 243. (6) Brusset, H.; Gillier-Pandraud, M. H.; Berdot, J.-L. Bull. Soc. Chim. Fr. 1967, 8, 2886. (7) Sandstrom, R. L.; Giess, E. A.; Gallagher, W. J.; Segmu¨ller, A.; Cooper, E. I.; Chisholm, M. F.; Gupta, A.; Shinole, S.; Laibowitz, R. B. Appl. Phys. Lett. 1988, 53, 1874. (8) Kebin, L.; Zhenzhong, Q.; Xijun, L.; Jingsheng, Z.; Yuheng, Z. Thin Solid Films 1997, 304 (1-2), 386. (9) Okazaki, H.; Arakawa, A.; Asahi, T.; Oda, O.; Aiki, K. Solid-State Electron. 1997, 41, 2p. 263. (10) Vasylechko, L.; Akselrud, L.; Morgenroth, W.; Bismayer, U.; Matkovskii, A.; Savytskii, D. J. Alloys Compd. 2000, 297, 46–52. (11) Senyshyn, A.; Vasylechko, L.; Knapp, M.; Bismayer, U.; Berkowski, M.; Matkovskii, A. J. Alloys Compd. 2004, 382, 84–91. (12) Galicka, K.; Szade, J.; Ruello, P.; Laffez, P.; Ratuszna, A. Appl. Surf. Sci. 2009, 255, 4355–4361. (13) Shiwu, G. Comput. Phys. Commun. 2003, 153, 190. (14) Karlheinz, S. J. Solid State Chem. 2003, 176, 319. (15) Azzam, R. M. A.; Bashara, N. B. Ellipsometry and polarized light; North-Holland Personal Library: Amsterdam, The Netherlands, 1987. (16) Wethkamp, T.; Wilmers, K.; Esser, N.; Richter, W.; Ambacher, O.; Angerer, H.; Jungk, G.; Johnson, R. L.; Cardona, M. Thin Solid Films 1998, 313-314, 745. (17) Hohenberg, P.; Kohn, W. Phys. ReV. B 1964, 136, 864. (18) Blaha, P.; Schwarz, K.; Madsen, G. K. H.; Kvasnicka, D.; Luitz, J. WIEN2K, “an Augmented Plane WaVe + Local orbitals program for calculating crystal properties”; Karlheinz Schwarz, Techn. Universitat: Wien, Austria, 2001; ISBN 3-9501031-1-2. (19) Anisimov, V. I.; Solvyev, I. V.; Korotin, M. A.; Czyzyk, M. T.; Sawatzky, C. A. Phys. ReV. B 1993, 48, 16929. (20) Liechtenstein, A. I.; Anisimov, V. I.; Zaanen, J. Phys. ReV. B 1995, 52, R5467. (21) Hufner, S.; Claessen, R.; Reinert, F.; Straub, Th.; Strocov, V. N.; Steiner, P. J. Electron Spectrosc. Relat. Phenom. 1999, 100, 191. Ahuja, R.; Auluck, S.; Johansson, B.; Khan, M. A. Phys. ReV. B 1994, 50, 2128. (22) Reshak, A. H.; Kityk, I. V.; Auluck, S. J. Chem. Phys. 2008, 129, 074706. (23) Tributsch, H. Z. Naturforsch., A 1977, 32, 972. (24) Reshak, A. H. Ph.D. thesis, Indian Institute of Technology-Rookee, India, 2005.
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